Itô's Theory of Excursion Point Processes and Its Developments

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Stochastic Processes and their Applications 120 (2010) 653–677 www.elsevier.com/locate/spa

Ito’sˆ theory of excursion point processes and its developments

Shinzo Watanabe

527-10, Chayacho, Higashiyama-ku, Kyoto, 605-0931, Japan Received 3 September 2009; accepted 25 September 2009 Available online 6 February 2010

Abstract

Ito’sˆ theory of excursion point processes is reviewed and the following topics are discussed: Application of the theory to one-dimensional diffusion processes on half-intervals satisfying Feller’s boundary conditions, and its multi-dimensional extension, i.e., the application of the theory to multi-dimensional diffusion processes satisfying Wentzell’s boundary conditions. c 2010 Elsevier B.V. All rights reserved.

MSC: 60J60; 60G55; 60J50; 60J65

Keywords: Poisson point processes; Excursion point processes; Feller’s boundary conditions; Wentzell’s boundary conditions

1. Introduction

In this paper, dedicated to the memory of Professor Kiyosi Ito,ˆ we shall review Ito’sˆ theory of excursion point processes and some of its further developments. As Itoˆ himself remembered in the Foreword of [13], this theory is a natural outgrowth of his joint work with Henry McKean [14,15] on one-dimensional diffusion processes: in order to understand better the class of diffusion processes on a linear interval satisfying Feller’s boundary conditions, Itoˆ introduced in [11,12] the notion of excursion point processes, which are point processes with values in a function space (i.e. path space). On the other hand, we know well that the notion of Poisson point processes has been introduced and it has played a key role in Levy–It´ oˆ theory on the structure of sample paths of Levy´ processes [8,10]. This class of Poisson point processes has been deﬁned by the size of

E-mail address: [email protected].

0304-4149/$ - see front matter c 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spa.2010.01.012 654 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 jumps of path functions, and so point processes take their values in a finite dimensional Euclidean space. However, the notion of point processes is defined generally with state spaces which can be rather arbitrary. Ito’sˆ idea in [11,12] is that, for a strong Markov process, the totality of excursions away from a recurrent state can be formulated as a stationary Poisson point process with values in a function space. This is indeed a breakthrough in the study of boundary problems for diffusion processes. We quote here a recollection of Ito’sˆ given in the Foreword of [13] in his study of the description of all possible extensions of a minimal diffusion up to the hitting time of the boundary point: After several years it became my habit to observe even finite dimensional facts from the infinite dimensional viewpoint. This habit led me to reduce the problem above to a Poisson point process with values in the space of excursions. A most typical and important example is the case of the Poisson point process of Brownian excursions which is treated, e.g., in Chap. III, Subsection 4.3 of [7], Chap. 6 of [17], Chap. VI, Section 8 of [21], Chap. XII of [20]. The Poisson point process of Brownian excursions is a correct and precise mathematical realization of the decomposition of Brownian sample functions into pieces called excursions. Once this is correctly formulated, not only Brownian sample functions but also their functionals, like local times, the maximum process, down-crossing numbers, occupation times, etc., can be recovered and represented in terms of the point process of Brownian excursions. Many facts concerning these functionals and their limit theorems can be deduced from such representations. Even if the facts considered thereby may not be new, this approach can often provide us with a deeper and better understanding of such facts and can produce sometimes a remarkable progress of the theory. Some such examples are the Williams formula for the occupation times on the positive half-line of Brownian paths which proves the arcsine law without appealing to the Feynman–Kac theorem, Levy’s´ down-crossing theorem, asymptotic laws of planar Brownian motion due to Pitman and Yor, among many others. As was stated above, the motivation of Itoˆ in the study of point processes with values in a function space originated from the joint work with McKean on analyzing and constructing path functions of linear diffusions satisfying the most general Feller boundary conditions. I could never forget those hot days in the summer of 1969, when Professor Itoˆ returned to his home in Kyoto, on vacation from his regular work at Aarhus, and gave us a series of lectures on excursion point processes (which was a source of [11]). I remember that he told us then that his theory was motivated by a question raised by Lamperti, “What are all possible boundary conditions at an exit boundary of a linear diffusion, typically the case of the boundary 0 of Feller’s diffusion 2 on [0, ∞) with the generator x d , which is a typical scaling limit of critical Galton–Watson dx2 branching processes?” We shall review this topic in Section 3. A main objective of the present paper is to extend Ito’sˆ idea to the same problem in multi- dimensional cases, in particular, to the problem of constructing and analyzing multi-dimensional diffusion processes in a domain with the boundary satisfying the most general Wentzell boundary conditions. This was a problem which, after the success of the Ito–McKeanˆ theory, was attacked by many people by various approaches. In Section 4, we shall show that Ito’sˆ approach by Poisson point processes of excursions can still be applied well to this problem. We shall discuss the problem in Section 4, going from a simpler case to more general cases. As we see there, the problem can be solved by determining and constructing two stochastic processes. The first one, denoted by (ξ(t)), is a process moving on the boundary. The second one, denoted by (A(t)), is an increasing process. A simple case of the problem is when the generator, which describes the behavior of the diffusion inside the domain before hitting the boundary, and the boundary condition, which S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 655 describes the behavior of the diffusion on the boundary, are both of constant coefficients. In this case, both processes (ξ(t)) and (A(t)) are Levy´ processes and they can be given directly by non- canonical representations of Levy–It´ oˆ type, as in Example 2.1 of Section 2, from two Poisson point processes with values in certain function spaces. So, in Section 4, we first set up these two Poisson point processes from the given data for the generator and the boundary condition. In more complicated cases of the data given by variable coefficients, we start again with the same two Poisson point processes with values in function spaces. In this case, however, the processes (ξ(t)) and (A(t)) are no longer Levy´ processes, but are semimartingales and should be determined as unique solutions of stochastic differential equations (SDE’s) of the jump type based on these Poisson point processes. So far, a SDE of the jump type has usually been considered only in the case when the SDE is based on a point process with values in a Euclidean space, as discussed, e.g., by Itoˆ [9] and Gihman and Skorohod [5]. In the present problem, we need and use essentially SDE’s based on Poisson point processes with values in function spaces.

2. Point processes and Poisson point processes

Let (U, BU ) be a measurable space. In this paper, we always assume that U is a standard Borel space in the sense of [19] and BU is the topological σ-ﬁeld on U (equivalently, U is a Lusin space in the sense of [2] and BU is the totality of Borel subsets of U). By a point function on U, we mean a mapping

p : (0, ∞) ⊃ Dp 3 t 7→ p(t) ∈ U where the domain Dp of the mapping p is a countable subset of (0, ∞). p deﬁnes a counting measure N p(ds, du) on (0, ∞) × U with σ-ﬁeld B(0, ∞) ⊗ BU via X N p((0, t] × A) = 1A(p(s)), t > 0, A ∈ BU . (2.1) s∈Dp,s≤t

Let ΠU be the totality of point functions on U and B(ΠU ) be the smallest σ-field on ΠU with respect to which the mappings ΠU 3 p 7→ N p((0, t] × A) ∈ {0, 1,..., ∞}, for each t > 0 and A ∈ BU , are all measurable. By a point process p, we mean a (ΠU , B(ΠU ))-valued random variable, i.e., a map p : Ω 3 ω → p(ω) ∈ ΠU , which is F/B(ΠU )-measurable, defined on a probability space (Ω, F, P). A point process p is called σ-finite if there exist An ∈ BU , S n = 1, 2,... , such that An ⊂ An+1, n An = U and N p((0, t]× An) < ∞ a.s. for all t > 0 and n. A general theory of point processes is developed in the framework of semimartingale theory (cf., e.g., [7,16,18]). In this paper, we treat mainly the notion of Poisson point processes.

Deﬁnition 2.1. A point process p is called a stationary Poisson point process if the following holds:

( l ) ! ( l ) X p X −λk E exp − λk N p((s, t] × Ak) |Fs = exp −(t − s) n(Ak)(1 − e ) (2.2) k=1 k=1 for every t > s ≥ 0, λk > 0 and Ak ∈ BU such that {Ak} are mutually disjoint, where n is a positive measure on (U, BU ) deﬁned by

n(A) = E{N p((0, 1] × A)}, A ∈ BU , 656 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677

p and Fs is the sub-σ-ﬁeld of F generated by the family {(r, p(r)); r ∈ Dp, r ≤ s} of random points in (0, s] × U.

Thus, the law of a stationary Poisson point process p is uniquely determined by the measure n. We call n the characteristic measure of p. When the underlying probability space is endowed with a ﬁltration (Ft ), a point process p is called an (Ft )-stationary Poisson point process if N p((0, t] × A) is Ft -measurable for every p t > 0 and A ∈ BU , and (2.2) holds with Fs replaced by Fs. Itoˆ [12] characterized a stationary Poisson point process as a point process having the following two properties: d (i) It is stationary in the sense that, for every t > 0, θt (p) = p, where θt (p) is the point process deﬁned by

Dθt (p) = {s; t + s ∈ Dp} and θt (p)(s) = p(t + s), s ∈ Dθt (p).

(ii) It is a renewal in the sense that, for every t > 0, the point process θt (p) is independent of the family {(s, p(s)); s ∈ Dp, s ≤ t} of random points in (0, t] × U. A characterization of an (Ft )-stationary Poisson point process in the semimartingale framework is as follows (cf. e.g. [7,16]): An (Ft )-adapted point process is an (Ft )-stationary Poisson point process if and only if its compensator Nbp((0, t] × A), A ∈ BU , is given by the deterministic measure t · n(A). A stationary Poisson point process is σ-finite if and only if its characteristic measure n is a σ-finite measure on (U, BU ). In the following, we treat only σ-finite stationary Poisson point processes. The most basic existence theorem is stated as follows: Given a σ-finite measure n on U, there exists a (law unique) stationary Poisson point process p with characteristic measure n.A standard construction of p from families of i.i.d. exponential times and i.i.d. U-valued random variables is explained in, e.g., [12,11,7]. Consider a point process p on a standard Borel space U. Let V be another standard Borel space and suppose we are given a Borel map T : U → V . Then a point process q on V is defined by setting Dq = Dp and q(t) = T (p(t)) for t ∈ Dp. We define q = T (p) and call it the image point process of p under the map T . When p is a stationary Poisson point process with characteristic measure n and if we suppose −1 that the image measure T (n) := n ◦ T of n under the map T is a σ-finite measure on (V, BV ), then it is obvious that T (p) is a stationary Poisson point process on V with characteristic measure T (n). Now we review the important notion of stochastic integrals based on Poisson point processes. This is usually treated in the framework of semimartingale theory, and so we assume that the underlying probability space (Ω, F, P) is endowed with a filtration (Ft ). Let p be an (Ft )-stationary Poisson point process with values in a standard Borel space U and with characteristic measure n. A (real or Rn-valued) function f (t, u, ω) defined on [0, ∞)×U ×Ω is called predictable if the map f : ([0, ∞)×Ω)×U 3 ((t, ω), u) 7→ f (t, u, ω) is P ⊗ BU -measurable, where P is the (Ft )-predictable σ-field on [0, ∞) × Ω. First, if a predictable function f (s, u, ω) satisfies the condition Z | f (s, u, ·)|N p(ds, du) < ∞ a.s., for every t > 0, (2.3) (0,t]×U S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 657 then we have, obviously, Z X f (s, u, ·)N p(ds, du) = f (s, p(s), ·), (2.4) (0,t]×U s∈Dp,s≤t the sum in the right-hand side (RHS) being absolutely convergent almost surely. If f (s, u, ω) satisfies the condition Z t Z ds E (| f (s, u, ·)|) n(du) < ∞, for every t > 0, (2.5) 0 U then (2.3) holds and we have the identity Z Z t Z E f (s, u, ·)N p(ds, du) = ds E( f (s, u, ·))n(du). (2.6) (0,t]×U 0 U Next, if a predictable function f (s, u, ω) satisfies the condition Z t Z ds E | f (s, u, ·)|2 n(du) < ∞, for every t > 0, (2.7) 0 U R · then the stochastic integral (0,t]×U f (s, u, )Nep(ds, du), which is also often denoted by R t+ R · 0 U f (s, u, )Nep(ds, du), is defined as a square-integrable (Ft )-martingale: here, we define, formally, Nep(ds, du) = N p(ds, du) − ds n(du) and the previous stochastic integral is defined as a limit of compensated sums (cf. e.g. [7] for details). If g(s, u, ω) has the same property as f (s, u, ω), then we have the identity Z Z E f (s, u, ·)Nep(ds, du) · g(s, u, ·)Nep(ds, du) (0,t]×U (0,t]×U Z t Z = ds E[ f (s, u, ·) · g(s, u, ·)]n(du). (2.8) 0 U (In the case of Rn-valued integrands, · is understood as the inner product.) Let U0 ∈ BU be such that

n(U \ U0) < ∞. (2.9) n · Then, for an R -valued predictable function f (s, u, ω), the function f (s, u, ω) 1U\U0 satisfies condition (2.3) because N p((0, t] × (U \ U0)) is finite a.s. for every t > 0. Suppose further that f (s, u, ω) satisfies the condition Z t Z ds E | f (s, u, ·)|2 n(du) < ∞, for every t > 0. (2.10) 0 U0 Under these conditions, both Z · f (s, u, )1U\U0 (u)N p(ds, du) (0,t]×U and Z · f (s, u, )1U0 (u)Nep(ds, du) (0,t]×U 658 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 are well-defined, and so an n-dimensional (Ft )-semimartingale ξ(t) is defined by Z = · ξ(t) f (s, u, )1U\U0 (u)N p(ds, du) (0,t]×U Z + · f (s, u, )1U0 (u)Nep(ds, du). (2.11) (0,t]×U We shall meet many examples of semimartingales represented in this form in Section 4.

Example 2.1 (Canonical and Non-canonical Levy–It´ oˆ Representations of Levy´ Processes). In the case when f (s, u, ·) is given by f (s, u, ·) = φ(u), where φ is a Borel map φ : U 3 u → φ(u) ∈ Rn satisfying Z |φ(u)|2n(du) < ∞, U0 the above (Ft )-semimartingale ξ(t) given by (2.11) is a stationary (Ft )-Levy´ process in the sense that, for every t > s ≥ 0, the increment ξ(t) − ξ(s) is independent of Fs and its law depends only on t − s. Thus, the very definition Z Z = + ξ(t) φ(u)1U\U0 (u)N p(ds, du) φ(u)1U0 (u)Nep(ds, du) (2.12) (0,t]×U (0,t]×U is a representation of the Levy–It´ oˆ type for ξ(t). Let Ue = {u ∈ U; |φ(u)| > 0}. If q is the image point process, under the map φ, of the point process p restricted to Ue, then q is an (Ft )- stationary Poisson point process with values in Rn \{0} and with characteristic measure ν, which is the image measure, under the map φ, of the measure n restricted to Ue. It is easy to verify that R ∧ | |2 ∞ Rn\{0}(1 x )ν(dx) < . It is also easy to deduce the following formula: Z Z ξ(t) = x1{|x|>1} Nq (ds, dx) + x1{|x|≤1} Neq (ds, dx) + bt (2.13) (0,t]×(Rn\{0}) (0,t]×(Rn\{0}) where b ∈ Rn is given by Z Z b = φ(u)n(du) − φ(u)n(du). (2.14) (U\U0)∩{|φ(u)|≤1} U0∩{|φ(u)|>1} This is the canonical Levy–It´ oˆ representation of the Levy´ process ξ(t) and the Levy´ measure in the Levy–Khinchin´ canonical form of the law of ξ(1) coincides with the characteristic measure ν of the point process q. When f (s, u, ·) is not a function of u only, the semimartingale ξ(t) given by (2.11) is no longer a Levy´ process. An important special case is when (2.11) defines a stochastic differential equation. An important result along these lines is summarized in the following theorem which will play a crucial role in Section 4. We set up, as above, a probability space (Ω, F, P) with a filtration (Ft ) and suppose an (Ft )-stationary Poisson point process p with values in a standard Borel space U and with characteristic measure n to be given. Let U0 ∈ BU be given as above such that

n(U \ U0) < ∞. (2.15) Suppose we are given the following: S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 659

= i n 3 7→ ∈ n ⊗ r (i) A Borel map σ (x) (σk (x)): R x σ (x) R R . (ii) A Borel map b(x) = (bi (x)): Rn 3 x 7→ b(x) ∈ Rn. (iii) A Borel map f (x, u) = ( f i (x, u)): Rn × U 3 (x, u) 7→ f (x, u) ∈ Rn. = i n ∈ n We set σk(x) (σk (x))i=1 R and assume that these functions satisfy (for some constant K > 0) r Z X 2 2 2 2 |σk(x)| + |b(x)| + | f (x, u)| n(du) ≤ K (1 + |x| ) (2.16) k=1 U0 and r Z X 2 2 2 |σk(x) − σk(y)| + |b(x) − b(y)| + | f (x, u) − f (y, u)| n(du) k=1 U0 ≤ K |x − y|2. (2.17)

k Theorem 2.1 (Cf. [7]). Given an r-dimensional (Ft )-Brownian motion (B (t)) and an F0- n n measurable and R -valued random variable ξ, the following SDE for an R -valued (Ft )- semimartingale (ξ(t)) such that ξ(0) = ξ has a pathwise unique solution: r Z t Z t X k ξ(t) = ξ(0) + σk(ξ(s))dB (s) + b(ξ(s))ds k=1 0 0 Z t+ Z + − · f (ξ(s ), u) 1U\U0 (u) N p(ds, du) 0 U Z t+ Z + − · f (ξ(s ), u) 1U0 (u) Nep(ds, du). (2.18) 0 U Finally, we give an example of Poisson point processes with values in a function space. As was explained in the Introduction, the study of such point processes has been a main motivation of Ito’sˆ works in [11,12]. Also, this example plays a fundamental role in the following sections.

Example 2.2 (Excursion Point Process of Brownian Excursions). Let B = (B(t)) be a reﬂecting Brownian motion on the half-line [0, ∞), with B(0) = 0, and let l(t) be the local time at 0 of B, that is, 1 Z t l(t) = lim 1[0,ε)(B(s))ds. ε→0 2ε 0 We introduce the following notation for path spaces: n W0 = ω :[0, ∞) 3 t 7→ ω(t) ∈ [0, ∞); continuous, ω(0) = 0, 0 < ∃σ (ω) < ∞ o such that ω(t) > 0 if t ∈ (0, σ (ω)) and ω(t) = 0 if t ≥ σ (ω) , (2.19) and, more generally, for x ∈ [0, ∞), n Wx = ω :[0, ∞) 3 t 7→ ω(t) ∈ [0, ∞); continuous, ω(0) = x, 0 < ∃σ (ω) < ∞ o such that ω(t) > 0 if t ∈ (0, σ (ω)) and ω(t) = 0 if t ≥ σ (ω) . (2.20) 660 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677

These spaces are all standard Borel spaces (cf. [21], p. 413). Let l−1(t) := A(t) be the right-continuous inverse of t 7→ l(t). Then t 7→ A(t) is strictly increasing and limt→∞ A(t) = ∞ almost surely. Then a random set in (0, ∞) is defined by D = {s > 0; A(s) > A(s−)}. For s ∈ D, we define a path ps ∈ W0 by setting B(A(s−) + t) − B(A(s−)), if 0 ≤ t < A(s) − A(s−), p (t) = (2.21) s 0, if t ≥ A(s) − A(s−), so σ (ps) = A(s) − A(s−). By setting Dp = D and p(s) = ps for s ∈ Dp, we have a point process p with values in the path space W0 and Itoˆ proved that this is a stationary Poisson point process. It is called the excursion point process of Brownian positive excursions. The excursion point process of Brownian negative excursions is defined similarly, from a reflecting Brownian motion on the negative half-line (−∞, 0]. For the excursion point process of Brownian positive excursions p, its characteristic measure n+ is a σ-finite measure on W0, which we call the Brownian (positive) excursion measure. This can be described in several different ways: a standard one is to define it as a Markovian measure associated with the law of absorbing Brownian motion on (0, ∞) having an entrance law from the origin. (An entrance law is determined up to a multiplicative constant, which corresponds to a normalization of the local time.) We give here a description in terms of BES(3)-diffusion process: BES(3) (denote it as BESa(3) when it starts at a) is a conservative diffusion on the half-line [0, ∞) with the generator 2 1 d + 2 d and the boundary 0 as an immediate entrance state. The transition probability 2 dx2 x dx density p(t, x, y), t > 0, x, y ∈ [0, ∞), with respect to the measure m(dy) = y2dy is given by  1  (g(t, x − y) − g(t, x + y)), if x, y > 0  xy p(t, x, y) = p(t, y, x) = 2  g(t, x), if x ≥ 0 and y = 0,  t 2 where g(t, x) = √1 exp − x . 2πt 2t Since p(t, x, y) is strictly positive and continuous on (0, ∞) × [0, ∞) × [0, ∞), we can 0 T = precisely define, for each fixed T > 0, the pinned or tied down BES (3)-diffusion X0 (X(t)), 0 ≤ t ≤ T , with the condition X(0) = 0 and X(T ) = 0 almost surely. It is obtained from BES0(3), (X(t)), by a Girsanov–Maruyama transformation (a change of the drift) defined by the following positive martingale (M(t)) with expectation 1: p(T − t, X(t), 0) M(t) = , 0 ≤ t < T. p(T, 0, 0) T ∩ { = } The law of X0 defines a probability measure on W0 σ (ω) T , which we denote by T P0 . Then, we have the following representation of the positive Brownian excursion measure (cf. e.g. [7], p. 125): For A ∈ B(W0), Z ∞ T dT n+(A) = P0 (A ∩ {ω; σ (ω) = T })√ . 0 2πT 3 0 There is another beautiful description of n+ due to Williams using two independent BES (3) 0 (cf. e.g. [7], p. 144): Let {X1(t)} and {X2(t)} be mutually independent BES (3)’s and, for a > 0, S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 661

a = { ; = } = a = a + a { } let σi inf t Xi (t) a , for i 1, 2. Set σ σ1 σ2 and deﬁne Y (t) by  a X1(t), if 0 ≤ t < σ1 ,  a a a Y (t) = X2(σ − t), if σ1 ≤ t < σ , 0, if t ≥ σ a.

Then, the path t 7→ Y (t) is in W0. We denote its law on W0 by Ra. Then it holds that, for A ∈ B(W0), Z ∞ da n (A) = R (A) . + a 2 0 a Starting from the Brownian excursion measure n+, which is an infinite but σ-finite measure on W0, we have a stationary Poisson point process p on W0. Then, from p, we can recover the path function of the reflecting Brownian motion t 7→ B(t) and its local time l(t) at 0 in the following way (cf. e.g., [7], Chap. III, Subsection 4.3). First, set Z X A(t) = σ (ω)N p(ds, dω) = σ (p(s)). (0,t]×W 0 s∈Dp,s≤t As we saw in Example 2.1 above, this is a stationary Levy´ process which is obviously increasing, so it is a subordinator. By rewriting it in canonical Levy–It´ oˆ form as we did above, we can identify it as a stable subordinator of exponent 1/2. t 7→ A(t) is strictly increasing and limt→∞ A(t) = ∞ a.s.. Then, setting A(0−) = 0, we see that, for every t ∈ [0, ∞), there is a unique s ∈ [0, ∞), denoted by s = l(t), such that A(s−) ≤ t ≤ A(s). If A(s−) < A(s), this implies that s ∈ Dp and we set B(t) = [p(s)](t − A((s−))). If A(s−) = A(s), we set B(t) = 0. We can identify t 7→ B(t) and t 7→ l(t) so defined with the reflecting Brownian sample function and its local time at 0. From a Poisson point process of Brownian positive excursions p+ and a Poisson point process of Brownian negative excursions p− which are mutually independent, their sum p = p+ + p−, defined in an obvious way, is a Poisson point process with values in the disjoint union W0 ∪ {−W0} and with the domain given by the disjoint union of Dp+ and Dp− . Then a sample function of Brownian motion on the real line starting at the origin and its local time at the origin can be defined from the point process p by the same procedure as above. If, in this construction, the point processes p+ and p− are modified so that, for a given constant p with 0 < p < 1, their characteristic measures are replaced by p · n+ and (1 − p) · n−, respectively, then we obtain a sample function of the skew Brownian motion with the skew parameter p.

3. Excursion point processes associated with diffusion processes on the half-line satisfying Feller’s boundary conditions

3.1

Here we give a typical example of Ito’sˆ excursion point process and thereby the construction of sample functions in the case of diffusion processes on the half-line [0, ∞) satisfying Feller’s boundary conditions. First, we consider the case when the diffusions are Brownian motions. Let X = (Xt , Px ), x ∈ [0, ∞), be a diffusion process (i.e., a strong Markov process with continuous paths) on the half-interval [0, ∞) which behaves as a Brownian motion inside (0, ∞); however, we allow a discontinuity of the sample path when it is on the boundary x = 0, and so 662 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 a jump may take place from the boundary to inside. Also, we attach an extra state point ∆ as the terminal point and allow some extinction (i.e., the jump to the terminal point ∆) when the process is on the boundary. As is shown in p. 194 of [14], such a diffusion is described by the following parameters: Three nonnegative numbers p1, p2, p3 and a nonnegative Borel measure p4(dx) on (0, ∞) subject to the normalization Z p1 + p2 + p3 + (1 ∧ x)p4(dx) = 1 (3.1) (0,∞) and also having the property that

if p1 < 1 and p2 + p3 = 0, then p4((0, 1]) = ∞. (3.2) 2 So the local generator G inside (0, ∞) is given by Gu(x) = 1 d u (x) and Feller’s boundary 2 dx2 condition is given by du Z p1u(0) + p3(Gu)(0) = p2 (0) + (u(x) − u(0))p4(dx) (3.3) dx (0,∞) for C2 functions u(x) on [0, ∞) which are bounded together with the derivatives. We always [ ∞ ∪ { } = 2 [ ∞ extend u to 0, ) ∆ so that u(∆) 0. We denote by Cb ( 0, )) the class of all such ∈ 2 [ ∞ functions. Set, for u Cb ( 0, )), du Z (Lu)(0) = −p1u(0) + p2 (0) − p3(Gu)(0) + (u(x) − u(0))p4(dx). (3.4) dx (0,∞) One way of stating that the local generator G and Feller’s boundary condition (3.3) determine the diffusion X is the following: X is a stochastic process on [0, ∞) ∪ {∆} with ∆ as a ∈ 2 [ ∞ trap uniquely characterized by the following property: for any u Cb ( 0, )), the process t 7→ [u(X(t)) − u(X(0))] is a semimartingale with the semimartingale decomposition given by Z t u(X(t)) − u(X(0)) = a martingale + (Gu)(X(s))ds + (Lu)(0)φ(t), 0 where φ(t) is a continuous increasing process such that Z t Z t φ(t) = 1{0}(X(s))dφ(s) and 1{0}(X(s))ds = p3φ(t). (3.5) 0 0

The constant p1 and the measure p4(dx) can be put together to deﬁne a nonnegative Borel measure p5(dx) on the extended half-line (0, ∞) ∪ {∆} so that

the restriction p5(dx)|(0,∞) = p4(dx) and 1{∆}(x)p5(dx) = p1δ{∆}(dx), (3.6) where δ{∆}(dx) is the unit measure at the point ∆. Obviously, conversely p1 and p4 can be recovered from p5. We can also rewrite (3.3) in the form du Z p3(Gu)(0) = p2 (0) + (u(x) − u(0))p5(dx), x ∈ [0, ∞) (3.7) dx (0,∞)∪{∆} ∈ 2 [ ∞ for any function u(x) Cb ( 0, )). The rewriting (3.7) of (3.3) corresponds, in a probabilistic language, to the identification of the absorption (i.e., the extinction or killing) with the jump to S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 663 the terminal state ∆. Note also that, when p1 = 1, the point 0 and ∆ must be identified so that our diffusion X is strong Markov and right-continuous. We are now going to construct the path functions of X by Ito’sˆ method; for a given starting point a ∈ [0, ∞), we construct the path function X a(t) of X such that X a(0) = a. We disregard the trivial case p1 = 1; in this case, X starting from inside is a Brownian motion before it hits the boundary 0 and is killed at once on hitting 0, so we need to identify 0 and ∆. Thus, we assume that p1 < 1 from now on. First of all, we take a sufficiently large probability space (Ω, F, P) with a filtration (Ft ) on which we can realize the following objects:

(i) A ﬁltration (Gt ) on Ω such that Gt ⊂ F0 for every t > 0 and a one-dimensional (Gt )- Brownian motion bBa = (bBa(t)) such that bBa(0) = a. (ii) A stationary (Ft )-Poisson point process p1 on W0 with characteristic measure n+, i.e. a Poisson point process of Brownian positive excursions (cf. Example 2.2). (iii) A stationary (Ft )-Poisson point process p2 with values in the product space [0, ∞) × W1 with characteristic measure given by the product measure p4(dx)× P1, P1 being the Wiener measure on W1 := {w ∈ C([0, ∞) → R); w(0) = 1}. −p t a (iv) An exponential time e such that P(e > t) = e 1 , t ≥ 0, which is independent of bB , p1 and p2.

Remark 3.1. In the same spirit as combining p1 and p4 to obtain p5, the above point process p2 and the exponential time e can be combined together to obtain a point process p3 with values in the product space ([0, ∞) ∪ {∆}) × W0. Though such an approach may be more elegant mathematically, we shall follow a more elementary and standard route here. We recall the notation for the spaces of paths of excursions. For x ∈ [0, ∞),

Wx = {ω :[0, ∞) 3 t 7→ ω(t) ∈ [0, ∞); continuous, ω(0) = x, 0 < ∃σ (ω) < ∞ such that ω(t) > 0 if t ∈ (0, σ (ω)) and ω(t) = 0 if t ≥ σ (ω)}. (3.8)

Note that, if x > 0, then for ω ∈ Wx , σ (ω) = min{s ≥ 0|ω(s) = 0}; while, for ω ∈ W0, ω(0) = 0 and σ (ω) = min{s > 0|ω(s) = 0} > 0. Also, we introduce the following notation: [ W++ = Wx . x>0 We introduce the following two operations on path spaces; here, 0 denotes the constant zero path: 0(t) ≡ 0. ≥ (c) : 3 7→ (c) ∈ ∪ { } (i) For each c 0, a map T1 W0 ω T1 (ω) W0 0 is deﬁned by  t cω , if c > 0 (c) = 2 T1 (ω)(t) c (3.9) 0, if c = 0.

(ii) A map T2 : (0, ∞) × W1 3 (x, w) 7→ T2((x, w)) ∈ W++ is deﬁned by  t 2 x · w , if 0 ≤ t < x m0(w) T2((x, w))(t) = x2 (3.10) 2 0, if t ≥ x m0(w) where m0(w) = inf{s > 0|w(s) = 0} for w ∈ W1. (Note that w(0) = 1 if w ∈ W1.) 664 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677

(p2) Let q1 be the image under the map T1 of the point process p1 and q2 be the image under the map T2 of the point process p2, so that q1 and q2 are stationary (Ft )-Poisson point processes with values in W0 and in W++, respectively, and with characteristic measures m1 and m2, respectively, given by

m1(A) = p2n+(A), A ∈ B(W0) and Z n h s i o m (A) = P w; s 7→ x · w ∧ m (w) ∈ A p ( x), A ∈ B W . 2 1 2 0 4 d ( ++) (0,∞) x

Then the images σ (q1) and σ (q2) of q1 and q2, respectively, under the map σ : W0 ∪ W++ 3 ω 7→ σ (ω) ∈ (0, ∞) are stationary (Ft )-Poisson point processes with values in (0, ∞), which we denote by π1 and π2, respectively. Their characteristic measures, denoted by ν1(dx) and ν2(dx), respectively, are given by Z dx y −y2/(2x) ν1(dx) = p2 √ , ν2(dx) = √ e p4(dy) dx. 2πx3 (0,∞) 2πx3 bBa = { ≥ | a = } Let m0 min s 0 bB (s) 0 . We set

bBa X X A(t) = m0 + σ (q1(s)) + σ (q2(s)) + p3t ∈ ; ≤ ∈ ; ≤ s Dq1 s t s Dq2 s t Z t+ Z Z t+ Z = bBa + + + m0 σ (ω)Nq1 (ds, dω) σ (ω)Nq2 (ds, dω) p3t 0 W0 0 W++ Z t+ Z Z t+ Z = bBa + + + m0 x Nπ1 (ds, dx) x Nπ2 (ds, dx) p3t. (3.11) 0 (0,∞) 0 (0,∞) R ∧ + R ∧ ∞ It is easy to see that (0,∞)(1 x)ν1(dx) (0,∞)(1 x)ν2(dx) < and hence (3.11) defines an (Ft )-subordinator (i.e., a right-continuous, increasing (Ft )-adapted process {A(t)} such that − ≤ A(t2) A(t1) is independent of Ft1 for every 0 t1 < t2). By our assumptions (3.1) and (3.2), we easily see that, with probability 1, t 7→ A(t) is strictly increasing and limt→∞ A(t) = ∞. Now we define the path t ∈ [0, ∞) 7→ X a(t) ∈ [0, ∞) ∪ {∆} as follows: ≤ ≤ bBa a = a If 0 t m0 , we set X (t) bB (t). bBa ≥ − = = If A(e) > t > m0 , then there exists a unique e s > 0 such that A(s ) A(s) t or A(s−) ≤ t < A(s). In the first case, we set X a(t) = 0. ∈ ∈ In the second case, it must happen that either s Dq1 or s Dq2 , and exactly one case can ∩ = ∅ ∈ a = occur with probability 1 because Dq1 Dq2 with probability 1. If s Dq1 , we set X (t) − − ∈ a = − − [q1(s)] (t A(s )), while if s Dq2 , we set X (t) [q2(s)] (t A(s )). Finally, if t ≥ A(e), we set X a(t) = ∆. In this way, {X a(t)} is completely defined and we can identify it as a Brownian motion on the half-line satisfying Feller’s boundary condition (3.3).

Remark 3.2. In the above construction, we have used the scaling property of the Wiener process in (3.9) and (3.10). This seemingly troublesome procedure, which we can spare in the one- dimensional case, will play a crucial role in its multi-dimensional extension, as we discuss in the next section. S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 665

3.2

We consider the general case in which the minimal diffusion bB is replaced by a more general Feller diffusion process bX. Ito’sˆ method, based on the Poisson point process of excursions, as we have discussed in the case of the process bB, still works. We shall only remark upon some necessary modifications. For simplicity, we assume that bX is given on the half-interval (0, ∞) = d d as a Feller diffusion with generator G m(dx) dx , so the canonical scale is the Euclidean scale: s(x) = x, and the speed measure is an everywhere positive Radon measure on (0, ∞). The boundary ∞ cannot be reached from inside and so we consider only the problem concerning the boundary 0. R { ]} = ∞ If (0,1) m (x, 1 dx , then the boundary 0 is natural and the process bX, starting from inside, cannot reach 0 in a finite time. So a possible extension of bX does not exist. R { ]} ∞ { ]} In the case (0,1) m (x, 1 dx < , the boundary 0 is regular or exit according as m (0, 1 is finite or infinite. In both cases, the excursion measure n+ exists as an infinite measure on the path space W0 and, as was shown by Fitzsimmons and Yano ([4]; cf. also [27]), it can be obtained by a time change from Ito’sˆ measure n+ of Brownian positive excursions. Hence, we have a stationary Poisson point process p on W0 with characteristic measure n+. Let ν1 be the characteristic measure of the point process σ[p] on (0, ∞) obtained as the image of p under the map σ : W0 3 ω 7→ σ (ω) ∈ (0, ∞). Then we can show that Z (1 ∧ x)ν1(dx) is finite or infinite according as the boundary 0 is regular or exit. (0,∞) Hence, in the regular case, the totality of extensions are described by the same Feller boundary = d du conditions as (3.3) in which, however, we understand that Gu m(dx) dx and the domain 2 [ ∞ Cb ( 0, )) needs to be modified. + In the exit case, we have, almost surely, R t R σ (ω)N (ds, dω) = ∞ for t > 0 so we 0 W0 p cannot define a subordinator. This fact implies that we must have p2 = 0 in the boundary condition, that is, a continuous entering inside from the boundary is impossible. Entering is possible only by jumping-in. The jumping-in measure p4(dx) is possible if and only if R ∞ R x { ]} ∧ ∞ 0 0 m (y, 1 dy 1 p4(dx) < ([11]; cf. also [28]). As we remarked in the Introduction, the study of all possible boundary conditions in this exit boundary case has been a main motivation for Itoˆ in his theory of excursion point processes.

4. The multi-dimensional case: Diffusions with Wentzell’s boundary conditions

We shall consider the multi-dimensional extension of the problem which we discussed in the previous section for one-dimensional half-intervals. A diffusion process in a multi-dimensional domain or manifold is described by Kolmogorov’s differential equations (i.e., a semielliptic second-order differential operator) and, when the boundary exists, its possible behavior on the boundary is determined by Wentzell’s boundary condition (cf. [26]). In the one-dimensional case, the problem was completely settled by Feller and by Itoˆ and McKean, for which we have discussed the construction of path functions in the previous section. After the success of the Ito–McKeanˆ theory, the problems concerning Wentzell’s boundary condition were studied by many people with various approaches: for example, in an analytical approach, Sato and Ueno [22] laid out a fundamental route and, by following it, Bony, Courrege` and Priouret [1] succeeded in constructing diffusions in a very general case; in a probabilistic approach, Ikeda [6] applied Ito’sˆ 666 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677

SDE in a two-dimensional case and Watanabe (cf. [7]) and El Karoui [3] applied the SDE method in a general case. Here, we shall show that Ito’sˆ method based on excursion point processes, as applied above in the one-dimensional case, still works well for this problem; such an approach has been discussed by Watanabe [25] (cf. also [7]), Takanobu [23] and Takanobu and Watanabe [24]. Let D = (Rd )+ = {x = (x1,..., xd ); xd ≥ 0} be the upper half-space of Rd , D◦ = {x ∈ D; xd > 0} be its interior, and so its boundary is a (d − 1)-dimensional hyperplane given by ∂ D = {x ∈ D; xd = 0}. We denote x ∈ D as x = (x¯, xd ), x¯ = (x1,..., xd−1), and so (x¯, 0) ∈ ∂ D. ∈ 2 Suppose we are given a second-order differential operator G on D, for u Cb (D) (the space of C2-functions on D, bounded and with bounded derivatives),

d d 1 X ∂2u X ∂u Gu(x) = ai j (x) (x) + bi (x) (x), (4.1) 2 ∂xi ∂x j ∂xi i, j=1 i=1 and Wentzell’s boundary condition: d−1 d−1 1 X ∂2u X ∂u Lu((x¯, 0)) = αi j ((x¯, 0)) ((x¯, 0)) + βi ((x¯, 0)) ((x¯, 0)) 2 ∂xi ∂x j ∂xi i, j=1 i=1 ∂u + µ((x¯, 0)) ((x¯, 0)) − ρ((x¯, 0)) · (Gu)((x¯, 0)) ∂xd Z d−1 ∂u + ¯ + − ¯ − X i ¯ u((x, 0) y) u((x, 0)) y i ((x, 0)) nx¯ (dy) {D\{0}}∩{|y|≤1} ∂x Z i=1 + {u((x¯, 0) + y) − u((x¯, 0))} nx¯ (dy), (4.2) {D\{0}}∩{|y|>1} where: (i) ai j (x) and bi (x)i, j = 1,..., d, are bounded Borel-measurable functions on D such that i j ji Pd i j i j d a (x) = a (x), i, j=1 a (x)ξ ξ ≥ 0 for all ξ ∈ R , (ii) αi j ((x¯, 0)), βi ((x¯, 0)), i, j = 1,..., d − 1, µ((x¯, 0)) and ρ((x¯, 0)) are bounded Borel- i j ¯ = ji ¯ Pd−1 i j ¯ i j ≥ measurable functions on ∂ D such that α ((x, 0)) α ((x, 0)), i, j=1 α ((x, 0))ξ ξ 0 for all ξ ∈ Rd−1, µ((x¯, 0)) ≥ 0 and ρ((x¯, 0)) ≥ 0, (iii) for each (x¯, 0) ∈ ∂ D, nx¯ (dy) is a nonnegative Radon measure on D \{0} such that Z 2 d |y ¯| + y nx¯ (dy) + nx¯ ({D \{0}} ∩ {|y| > 1}) < ∞, (4.3) {D\{0}}∩{|y|≤1} and the integral (4.3) is bounded in x¯.

Remark 4.1. We often denote a function f ((x¯, 0)) on the boundary simply by f (x¯), so we write αi j ((x¯, 0)) = αi j (x¯), βi ((x¯, 0)) = βi (x¯), and so on.

Remark 4.2. We have assumed that the coefﬁcients in G and L are bounded globally, which may seem too strong. However, by a standard localization argument, more general cases can be reduced to this case and so we do not hesitate to set such assumptions. Also, we treat here only conservative diffusions, although we know well how to treat more general cases; for the one-dimensional case, we treated the general case in the previous section.

Our objective is a diffusion process X = (X(t), Px ) on D determined by a pair of analytic data (G, L). S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 667

Deﬁnition 4.1. By a (G, L)-process X = (X(t), Px ), we mean a cadl` ag` process X(t) on D such = ∈ 2 ∈ [ ∞ 7→ that, under Px , X (0) x a.s., and, for every u Cb (D), the process t 0, ) u(X(t)) is a semimartingale with the semimartingale decomposition given by Z t Z t u(X(t)) = u(x) + a martingale + (Gu)(X(s))ds + (Lu)(X(s))dφ(s) (4.4) 0 0 where φ(s) is a continuous increasing process such that Z t Z t Z t 1∂ D(X (s))dφ(s) = φ(t) and 1∂ D(X(s))ds = ρ(X(s))dφ(s). (4.5) 0 0 0

It is well-known that, if we can show the uniqueness in law of a (G, L)-process, then it is a diffusion process. We shall extend our method based on excursion point processes given in the previous section to construct (G, L)-process and discuss its uniqueness in law.

= 1 4.1. The case of G 2 ∆ and Wentzell’s boundary operator L with constant coefﬁcients

The simplest case is when ai j (x), bi (x), αi j (x¯), βi (x¯), µ(x¯) and ρ(x¯) are all constants: ai j (x) = ai j , bi (x) = bi , αi j (x¯) = αi j , βi (x¯) = βi , µ(x¯) = µ, ρ(x¯) = ρ, and the measure i j i j i nx¯ (dy) := n(dy) is independent of x¯. We assume, for simplicity, that a = δ and b = 0, and = 1 ◦ so G 2 ∆; probabilistically, the minimal diffusion is a Brownian motion bB(t) in D before it hits the boundary ∂ D. So, we are given a symmetric, nonnegative deﬁnite (d−1)×(d−1)-matrix αi j , a (d − 1)-vector βi , nonnegative constants µ and ρ, and a nonnegative measure n(dy) on D \{0}.

Lemma 4.1. For some positive integer l, there exists a Borel map

g : (Rl )+ 3 u 7→ g(u) = (g¯(u), gd (u)) ∈ D (4.6) with g(0) = 0 which satisﬁes the following properties: (i) On the set {u ∈ (Rl )+; |u| ≤ 1}, the function g(u) is bounded. (ii) lim|u|→0 g(u) = 0. The image of the measure du on {u ∈ (Rl )+; |g(u)| > } under the map u 7→ y = g(u) (iii) |u|l+1 0 coincides with the measure n(dy) on D \{0}. Here, du is the l-dimensional Lebesgue measure. A proof is omitted. By (4.3), we have Z n o du |g ¯(u)|2 + gd (u) < ∞. l+1 (4.7) {0<|u|≤1}∩(Rl )+ |u| So, instead of the Wentzell boundary operator L of the type given by (4.2), we may start with one of the following form:

d−1 d−1 1 X ∂2u X ∂u Lu((x¯, 0)) = αi j ((x¯, 0)) + βi ((x¯, 0)) 2 ∂xi ∂x j ∂xi i, j=1 i=1 ∂u + µ ((x¯, 0)) − ρ · (Gu)((x¯, 0)) ∂xd 668 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 ( ) Z d−1 ∂u du + ¯ + − ¯ − X i ¯ u((x, 0) g(u)) u((x, 0)) g(u) ((x, 0)) + l + ∂xi |u|l 1 {0<|u|≤1}∩(R ) i=1 Z du + {u((x¯, ) + g(u)) − u((x¯, ))} . 0 0 l+1 (4.8) {|u|>1}∩(Rl )+ |u| We introduce several path spaces in D to describe excursions away from the boundary. We set W(D) = {w ∈ C([0, ∞) → D); w(t ∧ σ (w)) = w(t)} (4.9) where σ (w) = inf{t > 0; w(t) ∈ ∂ D}. (4.10) We set also W+(D) = {w ∈ W(D); σ (w) > 0}. (4.11) + + For x ∈ D, let Wx (D) (resp. Wx (D)) be the subclass of W(D) (resp, W (D)) consisting of all paths w such that w(0) = x. ◦ + + Note that, if x ∈ D , then Wx (D) = Wx (D), while, if x ∈ ∂ D, then Wx (D) = Wx (D)∪{x}, where x is the constant path at x: x(t) ≡ x. 1 We are now going to construct the ( 2 ∆, L)-diffusion process X; we shall define its path function X a(t) starting at a ∈ D, i.e., X a(0) = a. First of all, we take a sufficiently large probability space (Ω, F, P) with a filtration (Ft ) on which we can realize the following objects:

(i) A ﬁltration (Gt ) on Ω such that Gt ⊂ F0 for every t > 0 and a d-dimensional (Gt )-Brownian motion bBa = (bBa(t)) such that bBa(0) = a. (d−1) × (ii) A stationary (Ft )-Poisson point process p1 with values in the product space W0 W0, (d−1) × (d−1) = { ∈ [ ∞ → d−1 ; having characteristic measure P0 n+. Here, W0 w C( 0, ) R ) = } (d−1) − (d−1) w(0) 0 and P0 is the (d 1)-dimensional Wiener measure on W0 . The one- dimensional path space W0 and the measure n+ on W0 are the same as in Section 3.1; n+ is the excursion measure of one-dimensional positive Brownian excursions on W0 (cf. Example 2.2). (iii) A stationary (F )-Poisson point process p with values in the product space W d ×((Rl )+\ t 2 (0¯,1) {0}) with characteristic measure given by the product measure P(d) × du . Here, (0¯,1) |u|l+1 W d = {w ∈ C([0, ∞) → Rd ); w(0) = (0¯, 1)} (0¯ is the origin in Rd−1), and P(d) (0¯,1) (0¯,1) is d-dimensional Wiener measure on W d , u ∈ (Rl )+ \{0} and du is the l-dimensional (0¯,1) Lebesgue measure. ∼ d−1 (iv) A continuous (Ft )-Levy´ process (a Gaussian diffusion) η(t) on ∂ D = R with η(0) = 0, associated with the covariance matrix αi j , i, j = 1,..., d − 1, and the drift vector β˜i , i = 1,..., d − 1, given by Z du β˜i = βi − gi (u) · P(d) [σ (w) > h(u)−2] , (4.12) (0¯,1) l+1 (Rl )+∩{0<|u|≤1} |u| where, for w = (w,¯ wd ) ∈ W d , we set σ (w) = min{s ≥ 0; wd (s) = 0}. Here, the (0¯,1) function h(u) is deﬁned by (4.19) below. S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 669

We set up the point processes p1 and p2 as mutually independent: Then all the random elements a bB , p1, p2 and η are mutually independent. (d−1) × [ ¯ ] ¯ = ¯ ∈ An element of the product space W0 W0 is denoted by w, ω with w (w(t)) − W (d 1) and ω = (ω(t)) ∈ W . Also, an element in the product space W d × ((Rl )+ \{0}) is 0 0 (0¯,1) denoted by [w, u] with w = (w(t)) ∈ W d and u ∈ (Rl )+ \{0}. (0¯,1) We define the following mappings (in the following, when there is no fear of confusion, we often identify x¯ ∈ Rd−1 with (x¯, 0) ∈ ∂ D, for simplicity): : × { (d−1) × } 3 ¯ [ ¯ ] 7→ ¯ [ ¯ ] ∈ Φ ∂ D W0 W0 (x, w, ω ) Φ(x, w, ω ) Wx¯ (D) and d l + Ψ : ∂ D × {W × ((R ) \{0})} 3 (x¯, [w, u]) 7→ Ψ(x¯, [w, u]) ∈ W ¯ + (D). (0¯,1) (x,0) g(u) The definition of Φ is as follows: Φ(x¯, [w, ¯ ω]) = (Φ¯ (x¯, [w, ¯ ω]), Φd (x¯, [w, ¯ ω])),  t x¯ + µ ·w ¯ ∧ σ (ω) , in the case µ > 0, Φ¯ (x¯, [w, ¯ ω])(t) = µ2 (4.13) x¯, in the case µ = 0, and  t µ · ω , in the case µ > 0, Φd (x¯, [w, ¯ ω])(t) = µ2 (4.14) 0, in the case µ = 0. The definition of Ψ is as follows: Denoting Ψ(x¯, [w, u]) = (Ψ¯ (x¯, [w, u]), Ψ d (x¯, [w, u])) with w = (w,¯ wd ) and, remembering σ (w) = min{s ≥ 0; wd (s) = 0}, Ψ¯ (x¯, [w, u])(t)  t x¯ +g ¯(u) + gd (u) ·w ¯ ∧ σ (w) , in the case gd (u) > 0, = (gd (u))2 (4.15) x¯ +g ¯(u), in the case gd (u) = 0, and  t gd (u) · wd ∧ σ (w) , in the case gd (u) > 0, Ψ d (x¯, [w, u])(t) = (gd (u))2 (4.16) 0, in the case gd (u) = 0. Note that Φ(x¯, [w, ¯ ω])(0) = (x¯, 0) ∈ ∂ D, while Ψ(x¯, [w, u])(0) = (x¯, 0) + g(u), which is in D \{(x¯, 0)} when g(u) 6= 0. Then σ{Φ(x¯, [w, ¯ ω])} and σ {Ψ(x¯, [w, u])} are given, respectively, by µ2σ (ω) and d 2 ¯ [ ¯ ] [ ] (g (u)) σ (w), which are independent of x and we denote them by bσ ( w, ω ) and bσ ( w, u ), respectively. Then it is easy to deduce the following: Z [ ¯ ] · (d−1) ¯ σ ( w, ω ) 1{σ (ω)≤1} P0 (dw)n+(dω) (d−1)× b W0 W0 Z 2 = µ σ (ω) · 1{σ (ω)≤1}n+(dω) < ∞ (4.17) W0 670 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 and Z Z (d−1) ¯ = ∞ 1{σ (ω)>1} P0 (dw)n+(dω) 1{σ (ω)>1}n+(dω) < . (4.18) (d−1)× W0 W0 W0 We set h(u) = |g ¯(u)|2 + gd (u), u ∈ (Rl )+ \{0}, (4.19) Z (d) du σ ([w, u]) · 1{ ≤ −2 | |≤ } P (dw) (d) b σ (w) h(u) , 0< u 1 (0¯,1) l+1 W ×(Rl )+ |u| (0¯,1) Z (d) h d 2 −2i du = E (g (u)) σ (w); σ (w) ≤ h(u) · 1{ | |≤ } < ∞, (4.20) (0¯,1) 0< u 1 l+1 (Rl )+ |u| and Z (d) du 1{ −2 | | } P (dw) < ∞. (4.21) (d) σ (w)>h(u) or u >1 (0¯,1) l+1 W ×(Rl )+ |u| (0¯,1) For example, (4.20) and (4.21) can be verified easily from (4.7) if we note simple estimates like P(d) (σ (w) > h(u)−2) = O(h(u)) and E(d) (σ (w) ∧ h(u)−2) = O(h(u)−1) as |u| → 0. (0¯,1) (0¯,1) We define two ∂ D-valued functions: ϕ(x¯, [w, ¯ ω]) = Φ(x¯, [w, ¯ ω])(σ[Φ(x¯, [w, ¯ ω])]) −x ¯ (4.22) and ψ(x¯, [w, u]) = Ψ(x¯, [w, u])(σ[Ψ(x¯, [w, u])]) −x ¯. (4.23) Then we see immediately that

µ ·w(σ ¯ (ω)), in the case µ > 0, ϕ(x¯, [w, ¯ ω]) = (4.24) 0¯, in the case µ = 0, and, writing w = (w,¯ wd ),

g¯(u) + gd (u) ·w(σ ¯ (w)), in the case gd (u) > 0, ψ(x¯, [w, u]) = (4.25) g¯(u), in the case gd (u) = 0, so that both are independent of x¯. We denote them by ϕ([w, ¯ ω]) and ψ([w, u]), respectively. Then, we can easily obtain the following estimates: Z | [ ¯ ] |2 · (d−1) ¯ ∞ ϕ( w, ω ) 1{σ (ω)≤1} P0 (dw)n+(dω) < (4.26) (d−1)× W0 W0 and Z 2 (d) du |ψ([w, u])| · 1{ ≤ −2 | |≤ } P (dw) < ∞. (4.27) (d) σ (w) h(u) , 0< u 1 (0¯,1) l+1 W ×(Rl )+ |u| (0¯,1) ∼ d−1 We deﬁne an (Ft )-Levy´ process ξ = (ξ(t)) on ∂ D = R by setting Z t+ Z = a bBa + + [ ¯ ] · [ ¯ ] ξ(t) bB (m∂ D) η(t) ϕ( w, ω ) 1{σ (ω)≤1} Nep1 (ds, d( w, ω )) (d−1)× 0 W0 W0 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 671 Z t+ Z + [ ¯ ] · [ ¯ ] ϕ( w, ω ) 1{σ (ω)>1} Np1 (ds, d( w, ω )) (d−1)× 0 W0 W0 Z t+ Z + [ ] · − [ ] ψ( w, u ) 1{σ (w)≤h(u) 2, 0<|u|≤1} Nep2 (ds, d( w, u )) 0 W (d) ×(Rl )+ (0¯,1) Z t+ Z + [ ] · − [ ] ψ( w, u ) 1{σ (w)>h(u) 2 or |u|>1} Np2 (ds, d( w, u )), (4.28) 0 W (d) ×(Rl )+ (0¯,1) where

bBa a m∂ D = inf{s ≥ 0; bB (s) ∈ ∂ D}. (4.29) By (4.18), (4.21), (4.26) and (4.27), the right-hand side (RHS) of (4.28) is well-defined and it yields an (Ft )-Levy´ process. Also, we define an increasing (Ft )-Levy´ process A = (A(t)) by setting Z t+ Z = bBa + + [ ¯ ] [ ¯ ] A(t) m∂ D ρt σ ( w, ω )Np1 (ds, d( w, ω )) (d−1)× b 0 W0 W0 Z t+ Z + [ u] N s [ u] bσ ( w, ) p2 (d , d( w, )). (4.30) 0 W (d) ×((Rl )+\{0}) (0¯,1) By (4.17), (4.18), (4.20) and (4.21), the RHS of (4.30) is well-defined and yields an increasing [ ¯ ] = 2 [ ] = d 2 (Ft )-Levy´ process. Since bσ ( w, ω ) µ σ (ω) and bσ ( w, u ) (g (u)) σ (w), it is easy to see = ∞ + + R that limt→∞ A(t) a.s., if and only if µ ρ (Rl )+ 1{gd (u)>0}du > 0 (including the value infinity). It is also easy to see that t 7→ A(t) is strictly increasing a.s., if and only if the following condition(H1) is satisfied.

(H1) At least one of the following three conditions is satisﬁed: Z du µ > , ρ > , 1 d = ∞. (i) 0 (ii) 0 (iii) {0<|u|≤1, g (u)>0} l+1 (Rl )+ |u|

Note that we have limt→∞ A(t) = ∞ a.s., when (H1) is satisfied. In the following, we assume that(H1) is satisfied. Now we can define the path function X a(t). Setting A(0−) = 0, the following holds with probability 1: For any given t ∈ [0, ∞), there exists a unique s ∈ [0, ∞), denoted by s = φ(t), such that A(s−) ≤ t ≤ A(s). a If s = 0, then this implies that 0 ≤ t ≤ m bB and we set X a(t) = bBa(t). − ∈ ∪ If s > 0 and if A(s) > A(s ), then this implies that s Dp1 Dp2 . By the independence of ∩ = ∅ ∈ ∈ p1 and p2, we have Dp1 Dp2 a.s., so that only one case, s Dp1 or s Dp2 , can occur. In the first case, we set a X (t) = Φ(ξ(s−), p1(s))(t − A(s−)). In the second case, we set a X (t) = Ψ(ξ(s−), p2(s))(t − A(s−)). s A s = A s− s ∈ s = If > 0 and if ( ) ( ), then this implies that either Dp2 and bσ (p2( )) 0, or 6∈ ∪ s Dp1 Dp2 . 672 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677

In the ﬁrst case, we set a X (t) = ξ(s)(=Ψ(ξ(s−), p2(s))(0)) = ξ(s−) +g ¯(us) where u denotes the “u-component” of p (s) ∈ W (d) × ((Rl )+ \{0}). s 2 (0¯,1) In the second case, we have ξ(s) = ξ(s−) and we set X a(t) = ξ(s). In this way, we have completely deﬁned the path function {X a(t)}. We can identify this as a (G, L)-process and show its uniqueness in law.

= 1 4.2. The case of G 2 ∆ and with Wentzell’s boundary operator L of variable coefﬁcients

= 1 Here, we consider (G, L)-diffusion processes with G 2 ∆ as in the previous subsection but the boundary operator L has variable coefficients. So we treat the case of L given in the same form as (4.8) in which, however, αi j , βi , µ, ρ, and g(u) = (g¯(u), gd (u)) may depend on the boundary points x¯ ∈ ∂ D, so that they are replaced by αi j (x¯), βi (x¯), µ(x¯), ρ(x¯), g(x¯, u) = (g¯(x¯, u), gd (x¯, u)). Of course, these functions satisfy, at each fixed x¯ ∈ ∂ D, the same conditions as were given in the previous section and we assume, for simplicity, that these functions are bounded on ∂ D. As for g(x¯, u) = (g¯(x¯, u), gd (x¯, u)), we assume, more precisely, that there exists a positive function h(u), of u ∈ (Rl )+, having the property that h(0) = 0, lim|u|→0 h(u) = 0, bounded on {0 < |u| ≤ 1} and Z du h(u) < ∞, l+1 (4.31) (Rl )+∩{0<|u|≤1} |u| which satisfies |g ¯(x¯, u)|2 + gd (x¯, u) ≤ h(u) if x¯ ∈ ∂ D and 0 < |u| ≤ 1. (4.32) Now we define the mappings : × { (d−1) × } → Φ ∂ D W0 W0 W(D) and Ψ : ∂ D × {W d × ((Rl )+ \{0})} → W(D) (0¯,1) in the same way as (4.13) and (4.14) and as (4.15) and (4.16), respectively, in which, however, we replace µ and g(u) = (g¯(u), gd (u)) by µ(x¯) and g(x¯, u) = (g¯(x¯, u), gd (x¯, u)), respectively. Then, ¯ [ ¯ ] := { ¯ [ ¯ ] } ¯ [ ] := { ¯ [ ] } bσ (x, w, ω ) σ Φ(x, w, ω ) , and bσ (x, w, u ) σ Ψ(x, w, u ) now depend on x¯. Also, ∂ D-valued functions ϕ(x¯, [w, ¯ ω]) and ψ(x¯, [w, u]) are defined by (4.22) and (4.23), respectively, which now really depend on x¯. Then, (4.24) and (4.25) hold in which, however, µ and g(u) = (g¯(u), gd (u)) are replaced by µ(x¯) and g(x¯, u) = (g¯(x¯, u), gd (x¯, u)), respectively. Now, (4.18) and (4.21) hold and estimates like (4.17), (4.20), (4.26) and (4.27) still hold by modifying the dependence in x¯; in these estimates, the integrals in the LHS’s are uniformly bounded in x¯. S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 673

The path function of a (G, L)-diffusion can be defined in the same way as in the previous section from a process (ξ(t)) on the boundary ∂ D and an increasing process (A(t)). However, they are no longer Levy´ processes. Eq. (4.28) for the process (ξ(t)) now turns out to be a SDE of jump type (cf. (4.36) below). We add further sufficient conditions on the coefficients of L so that the SDE has a pathwise unique solution (in the following, K denotes some positive constant): i ¯ = − = (1) There exist bounded functions τk (x), i 1,..., d 1, k 1,..., r, such that r i j ¯ = X i ¯ j ¯ α (x) τk (x)τk (x) k=1 and i i |τk (x¯) − τk (y¯)| ≤ K |x ¯ −y ¯|. (2) |βi (x¯) − βi (y¯)| ≤ K |x ¯ −y ¯|. (3) |µ(x¯) − µ(y¯)| ≤ K |x ¯ −y ¯|. (4) For every u ∈ (Rl )+ such that 0 < |u| ≤ 1, |g ¯(x¯, u) −g ¯(y¯, u)|2 ≤ K · h(u)|x ¯ −y ¯|2, and |gd (x¯, u) − gd (y¯, u)| ≤ K · h(u)|x ¯ −y ¯|.

We can deduce the following estimates from (3) and (4): Z | ¯ [ ¯ ] − ¯ [ ¯ ] |2 · (d−1) ¯ ϕ(x, w, ω ) ϕ(y, w, ω ) 1{σ (ω)≤1} P0 (dw)n+(dω) (d−1)× W0 W0 ≤ K |x ¯ −y ¯|2 (4.33) and Z 2 (d) du |ψ(x¯, [w, u]) − ψ(y¯, [w, u])| · 1{ ≤ −2 | |≤ } P (dw) (d) σ (w) h(u) , 0< u 1 (0¯,1) l+1 W ×(Rl )+ |u| (0¯,1) ≤ K |x ¯ −y ¯|2. (4.34) a = 1 We are now going to construct a path function (X (t)) of the (G 2 ∆, L)-diffusion process starting at a ∈ D. First of all we set up the following on some probability space with a ﬁltration (Ft ).

(iii) A stationary (F )-Poisson point process p with values in the product space W d ×((Rl )+\ t 2 (0¯,1) {0}) with characteristic measure given by the product measure P(d) × du . Here, (0¯,1) |u|l+1 W d = {w ∈ C([0, ∞) → Rd ); w(0) = (0¯, 1)} (0¯ is the origin in Rd−1), and P(d) (0¯,1) (0¯,1) is d-dimensional Wiener measure on W d , u ∈ (Rl )+ \{0} and du is the l-dimensional (0¯,1) Lebesgue measure. = k r = (iv) An r-dimensional (Ft )-Brownian motion B (B (t))k=1 with B(0) 0. Define the functions β˜i (x¯), i = 1,..., d − 1, on ∂ D by Z du β˜i (x¯) = βi − gi (x¯, u) · P(d) [σ (w) > h(u)−2] . (4.35) (0¯,1) l+1 (Rl )+∩{0<|u|≤1} |u| ˜i It is easy to see that β (x¯) is also Lipschitz continuous. Define the ∂ D-valued functions τ¯k(x¯), k = 1,..., r, and β(ˇ x¯) by setting ¯ ¯ = i ¯ d−1 ˇ ¯ = ˜i ¯ d−1 τk(x) (τk (x))i=1 , β(x) (β (x))i=1 . Consider the following SDE of jump type: r Z t Z t a bBa X k ˇ ξ(t) = bB (m∂ D) + τ¯k(ξ(s))dB (s) + β(ξ(s))ds 0 0 Z t+ Z k=1 + ϕ(ξ(s−), [w, ¯ ω]) · 1{ ≤ } N (ds, d([w, ¯ ω])) − σ (ω) 1 ep1 0 W (d 1)×W Z t+ Z 0 0 + ϕ(ξ(s−), [w, ¯ ω]) · 1{ } N (ds, d([w, ¯ ω])) − σ (ω)>1 p1 0 W (d 1)×W Z t+ Z 0 0 + − [ ] · − [ ] ψ(ξ(s ), w, u ) 1{σ (w)≤h(u) 2, 0<|u|≤1} Nep2 (ds, d( w, u )) 0 W (d) ×(Rl )+ (0¯,1) Z t+ Z + − [ ] · − [ ] ψ(ξ(s ), w, u ) 1{σ (w)>h(u) 2 or |u|>1} Np2 (ds, d( w, u )). (4.36) 0 W (d) ×(Rl )+ (0¯,1) ˇ By the estimates (4.33) and (4.34) combined with the Lipschitz continuity of τ¯k and β, we can apply Theorem 2.1 to conclude that ξ(t) is determined uniquely as the pathwise unique solution of (4.36). Then A(t) can be defined by Z t Z t+ Z bBa A(t) = m + ρ(ξ(s))ds + σ (ξ(s−), [w, ¯ ω])Np (ds, d([w, ¯ ω])) ∂ D (d−1) b 1 0 0 W ×W0 Z t+ Z 0 + s− [ u] N s [ u] bσ (ξ( ), w, ) p2 (d , d( w, )). (4.37) 0 W (d) ×((Rl )+\{0}) (0¯,1)

We assume that the following condition (H2) is satisﬁed:

(H2) For every x¯ ∈ ∂ D, one of the following three conditions holds: Z du µ(x¯) > , ρ(x¯) > , 1 d = ∞. (i) 0 (ii) 0 (iii) {0<|u|≤1, g (x¯,u)>0} l+1 (Rl )+ |u|

Then we can conclude that, with probability 1, t 7→ A(t) is strictly increasing and limt→∞ A(t) = ∞. Now, X a(t) can be constructed in exactly the same way as in the previous subsection. S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677 675

4.3. The case of G being a general second-order elliptic differential operator

2 We consider the case of Gu(x) = 1 Pd ai j (x) ∂ u (x) + Pd bi (x) ∂u (x). We assume 2 i, j=1 ∂xi ∂x j i=1 ∂xi i that there exist bounded functions θq (x), i = 1,..., d, q = 1,..., m, such that m i j X i j a (x) = θq (x)θq (x). q=1 We also assume the Lipschitz conditions: i i i i |θq (x) − θq (y)| + |b (x) − b (y)| ≤ K |x − y|. We assume further that add (x) = 1 on D and

d d θq (x) = 0, q = 1,..., m − 1 and θm(x) = 1. (4.38) This assumption is not so restrictive; if add (x) > 0 everywhere on D, then, at least locally, the problem can be reduced to this case by a method of time change or a transformation of coordinates. As for the boundary operator L described by functions αi j (x¯), βi (x¯), µ(x¯), ρ(x¯) and g(x¯, u) = (g¯(x¯, u), gd (x¯, u)), we make all the same assumptions as we did in the previous subsection. We shall apply the same method as in the previous section: A necessary modification is − that, instead of the product spaces W (d 1) × W and W d × ((Rl )+ \{0}), we take spaces 0 0 (0¯,1) − W (m 1) × W and W m × ((Rl )+ \{0}), respectively, whose elements are denoted by the same 0 0 (0¯,1) − notation [w, ¯ ω] and [w, u] as above, with w¯ ∈ W (m 1), w ∈ W m and 0¯ is the origin in Rm−1. 0 (0¯,1) We define the mappings : × { (m−1) × } 3 ¯ [ ¯ ] 7→ ¯ [ ¯ ] ∈ Φ ∂ D W0 W0 (x, w, ω ) Φ(x, w, ω ) Wx¯ (D) and m l + Ψ : ∂ D × {W × ((R ) \{0})} 3 (x¯, [w, u]) 7→ Ψ(x¯, [w, u]) ∈ W ¯ + (D). (0¯,1) (x,0) g(u) To define them, we first consider the following d-dimensional SDE for Y (t) = (Y i (t)) on the { (m−1) × (m−1) × +} ¯ = 1 d−1 measure space W0 W0, P0 n : for given x (x ,..., x ), m−1 Z t i i X i q Y (t) = x + c θq (Y (s))dw (s) q=1 0 Z t Z t i 2 i + c θm(Y (s))dω(s) + c b (Y (s))ds, i = 1,..., d − 1, 0 0 Y d (t) = cω(t), (4.39) where c is a given nonnegative constant. Since n+ is the Brownian excursion measure, the stochastic integral for dω(s) can be defined as in ordinary Itoˆ calculus and the above SDE has a pathwise unique solution. We denote this solution as Y (c) = (Y (c)(t;x ¯, [w, ¯ ω])). Next, we consider the following d-dimensional SDE for Z(t) = (Zi (t)) on the Wiener space {W (m) , P(m) }: for given x¯ = (x1,..., xd−1), (0¯,1) (0¯,1) 676 S. Watanabe / Stochastic Processes and their Applications 120 (2010) 653–677

m Z t Z t i i X i q 2 i Z (t) = x + c θq (Z(s))dw (s) + c b (Z(s))ds, i = 1,..., d − 1, q=1 0 0 Z d (t) = cwm(t), (4.40) where c is a given nonnegative constant. We denote its pathwise unique solution by Z (c) = (Z (c)(t;x ¯, w)). Now the deﬁnition of Φ(x¯, [w, ¯ ω]) is as follows:  t Y (µ(x¯)) ∧ σ (ω);x ¯, [w, ¯ ω] , in the case µ (x¯) > 0, Φ(x¯, [w, ¯ ω])(t) = µ2(x¯) (4.41) (x¯, 0), in the case µ(x¯) = 0. The deﬁnition of Ψ(x¯, [w, u]) is as follows: letting σ (w) = inf{s ≥ 0; wm(s) = 0}, Ψ(x¯, [w, u])(t)

 d t Z (g (x¯,u)) ∧ σ (w);x ¯, w , in the case gd (x¯, u) > 0, = (gd )2(x¯, u) (4.42) (x¯ +g ¯(x¯, u), 0), in the case gd (x¯, u) = 0. Note that Φ(x¯, [w, ¯ ω])(0) = (x¯, 0) and Ψ(x¯, [w, u])(0) = (x¯, 0) + g(x¯, u). ¯ [ ¯ ] ¯ [ ] ¯ [ ¯ ] ¯ [ ] Using these Φ and Ψ, bσ (x, w, ω ), bσ (x, w, u ), ϕ(x, w, ω ) and ψ(x, w, u ) are defined in the same way as above and the construction of the path function X a(t) can be carried out in the same way under the same condition (H2). A necessary modification is just that the d- a dimensional (Gt )-Brownian motion (bB (t)), which was used to construct the part of the diffusion a before hitting the boundary, should be replaced by a (Gt )-measurable continuous process bX (t) on Rd which starts at a ∈ D. This process is obtained as the pathwise unique solution of the SDE m Z t Z t i i X i q i bX (t) = a + θq (bX(s))dB (s) + b (bX(s))ds, i = 1,..., d, q=1 0 0 q where (B (t)) is an m-dimensional (Gt )-Brownian motion which we first set up on the probability space. Now, we can identify this process (X a(t)) as the unique (G, L)-diffusion in law under the conditions on the coefficients that we imposed above. The details appear in [24], though some modifications should be made which we omit in this paper.

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